the angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 5.8m away from the wall. The length of the ladder is?
To find the length of the ladder, we can use trigonometry - specifically, the sine function.
Let's call the length of the ladder "L".
We are given that the angle of elevation of the ladder is 60° and the distance from the foot of the ladder to the wall is 5.8m.
In a right-angled triangle, the sine of an angle is equal to the opposite side divided by the hypotenuse.
In this case, the ladder is the hypotenuse, and the opposite side to the angle of elevation is the height of the ladder on the wall.
Therefore, we can use the equation:
sin(60°) = height of ladder on the wall / L
To find the height of the ladder on the wall, we need to calculate sin(60°). The sine of 60° is √3/2.
So, we can rewrite the equation as:
√3/2 = height of ladder on the wall / L
Next, we need to isolate L (the length of the ladder) by cross-multiplying:
√3/2 * L = height of ladder on the wall
Now we can solve for L:
L = (2 * height of ladder on the wall) / √3
To find the height of the ladder on the wall, we need to use the distance from the foot of the ladder to the wall (5.8m) and the relationship between the height and distance:
tan(60°) = height of ladder on the wall / 5.8m
The tangent of 60° is √3.
So, we can rewrite the equation as:
√3 = height of ladder on the wall / 5.8m
Next, we need to isolate the height:
height of ladder on the wall = √3 * 5.8m
Now, let's substitute this value back into the equation for L:
L = (2 * (√3 * 5.8m)) / √3
Simplifying this expression, we get:
L = (2 * √3 * 5.8m) / √3
L = 11.6m
Therefore, the length of the ladder is approximately 11.6 meters.